A flow around a square cylinder at the Reynolds number \({Re}_D=300\)  is then used for our presentation with the machine learning-based PIV velocity estimator [1]} in section .
The training dataset is prepared by a DNS, which has been verified against Franke et al. [2]} and Robichaux et al. [3]}, with numerically solving the incompressible Navier–Stokes equations with a penalization term [4]},
\(&{\mathbf {\nabla }} \cdot {\mathbf {u}}=0, \\&{\partial _t {\mathbf {u}}} + {\mathbf {\nabla }} \cdot \left({\mathbf {u}}{\mathbf {u}}\right)=-{\mathbf {\nabla }} p + \frac{1}{{ Re}_D}{\mathbf {\nabla }}^2\mathbf {u}+\lambda \chi \left({\mathbf {u}}_b-{\mathbf {u}}\right),\\ \nonumber \) 
